Commentary invited by editors of Scientific American

Keep It Simple, Stupid: Math Doesn’t Have to Be “Complex”

Photo: Wallpoper via Wikimedia Commons

When we go to the cinema, we expect certain things of big-screen scientists. Most of us will get annoyed if a film gets its basic facts wrong, for example. Directors are aware of this, so generally try to avoid schoolboy errors. After all, nobody wants to get pulled up on accuracy by a young child.

Yet according to David Kirby, in his book Lab Coats in Hollywood, filmmakers believe that some inaccuracies are necessary. They think that without certain (incorrect) stereotypes – like bubbling test tubes in a lab – the audience won’t buy into the story.

The “inaccurate-but-necessary stereotype” crops up in other media too. When researchers recently looked at how plants regulate their starch consumption, newspapers reported that the plants were using “complex maths”. While searching for food, it seems bees solve complex maths problems too. And if you want to create codes, you’ll need “complicated maths”.

We also get stories about people who’ve strung together calculations, creating “complex algorithms” that can guide passengers to taxis, help campers put up a tent or point gamblers towards correct results.

This is not to criticise the research behind these stories. There are many examples of mathematical models and algorithms that are innovative and even ingenious. They are just rarely described as such. When maths is involved, the temptation is to open the thesaurus at “complex”.

Why does this happen? One reason is the language barrier. Mathematical notation can look a bit alien (sometimes even to mathematicians). But almost anything will appear complicated when translated into an unknown language. It doesn’t necessarily mean the symbols represent a difficult concept.

Take Rolle’s Theorem. Using mathematical notation, this can be written as follows:

Lots of symbols, but the basic idea is fairly straightforward. Say we have two points – call them a and b – on a horizontal line. Then we draw a smooth line (i.e. no jagged zig-zags) between them without taking the pen off the paper. It might look like this:

Rolle’s Theorem states that there will be at least one point between a and b where the line above (or below) that point has a slope of zero:

That’s all there is to it. A bit underwhelming? Perhaps. Complicated? Hardly.

Symbols contribute to maths’ reputation for complexity, but it’s more than just a language problem. Mathematical results can also be counter-intuitive. Who wouldn’t be surprised to learn that in a room of 23 people, there is a 50% chance two of them share a birthday? When unexpected results like these are pulled apart, however, there is often a neatly formed insight lurking inside. Despite the popular image of Jackson Pollock-esque blackboards, mathematicians generally prefer elegant solutions to messy, confusing ones.

Some might argue such scientific stereotypes aren’t a problem. Yes the makers of Jurassic Park placed bubbling test tubes next to real molecular biology equipment, but did it really ruin the story?

Errant test tubes are one thing; presenting a subject as complex and impenetrable is far more problematic. First, it makes people suspicious of mathematical ideas. If we don’t understand something, our instinct is often not to trust it. As a result, useful work can get lumped together with the bad.

Whether in finance or science, mathematical models are an increasingly important part of modern life. While we should always keep a healthy scepticism about new models, we should direct it at the specific quirks and caveats of these models. If we presume all mathematical approaches come from the same complicated, suspicious heap, then harmful models will escape incisive criticism and useful ones will be ignored.

Blanket suspicion isn’t the only problem. Equally concerning is when people put too much faith in seductive symbols and scribbles. On several occasions, I’ve heard people call a maths PhD thesis brilliant because no one could understand it. To them, complicated meant clever.

Some people like to encourage this view. If you work in a competitive industry, there are benefits to having others in awe of your work. It can give you power, and help you avoid criticism. But when your marvellous theory runs into trouble, admiration can quickly turn to anger, and we arrive back at previous problem.

Mathematics is not some kind of opaque, untrustworthy black magic. Nor is it an infallible solution to every dilemma. It’s just a set of ideas, which can help us understand our world. As with any subject, some bits are difficult and some are surprisingly easy. But in the words of mathematician Stan Gudder, “The essence of mathematics is not to make simple things complicated, but to make complicated things simple.”

Adam Kucharski does not work for, consult to, own shares in or receive funding from any company or organisation that would benefit from this article, and has no relevant affiliations.

About the Author: Adam Kucharski is a postdoctoral researcher in mathematical epidemiology at Imperial College London, UK.

The views expressed are those of the author and are not necessarily those of Scientific American.

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1. 1. m 9:13 pm 08/29/2013

Nice.

It would be great if every mathematical symbol had an explanation with it. Just like computer programmers have to do, but I suppose that would be a tautology, as the maths is the explanation and the process all in one.

2. 2. m 9:19 pm 08/29/2013

Are there any programs written by someone, that takes in all these mathematical symbols (rules) and translates them to the more complicated English language equivalent.

So that an entire mathematical theorem can be put into it and it spits out what was explained here?

There are a few really complicated theorems Id like translated…

3. 3. jtdwyer 6:47 am 08/30/2013

Yes, nice article (especially for those of us who can encode complex processes into a complex series of arcane computer instructions, yet abhor math for its seemingly indecipherable symbols).

What those who are competent in math can fail to realize is: calculations that are correct within some certain context does not ensure that the context has correctly been applied. In particular, I’d really like to be able to demonstrate to physicists that Keplerian dynamics do not properly apply to the mass distributions of spiral galaxies, but words alone seem to be inadequate. See http://fqxi.org/data/essay-contest-files/Dwyer_FQXi_2012__Questionin_1.pdf for example.

So I wholeheartedly agree with ‘m’ that a math -> English translator would be very useful. I certainly also think that an English -> math translator would also be extremely useful! There seem to be several (but not all) scientific disciplines in which only mathematical formalism can be relied upon! In any case, maths seem to always be more impressive than mere words!

4. 4. KBCash 7:50 am 08/30/2013

Sorry, some of these don’t easily translate into English. That is the point of the symbols. How simple do you want it, do I have to explain what “3″ means?

This is why we learn the language. Mathematics, like French or Russian, is a language, and if you want to know what it means, you need to speak math.

http://kblakecash.wordpress.com/2013/08/29/chemistry/

5. 5. Logicoco 8:26 am 08/30/2013

Great message!

As a non-native English phd student on the fringe between economics and cultural studies, I experienced how cultural studies use language to make things terribly complicated presumably to make things look more sophisticated. Sometimes I read papers and think to myself what a lengthy complicated piece of writing for saying such trivial common sense things.

And if I attempt to make these things look simple using simple words, I get feedback from examiners that it is not “scholarly writing”. This makes me mad.

I would even go as far as to say that some academic scholars live from this practice because there is not much left if you take their sophisticated language away.

6. 6. leeml 4:14 pm 08/30/2013

(1) Technical symbols assure accuracy. Consider the Rolle’s Theorem example. Drawing a curve without lifting your pencil off the paper won’t necessarily produce a curve for which Rolle’s Theorem is true. Perhaps it produces the graph of a continuous function, but it may not produce one that has a derivative everywhere. The canonical example is something like the absolute value function on the interval from -1 to 1. There is no point at which Rolle’s Theorem is satisfied because there is no point at which the curve has a derivative of 0. The stated conditions of continuity on the closed interval and differentiability on its interior are necessary. Of course, the author chose a particularly symbol-rich way of stating these conditions. Most texts would state it as: If f is continuous on [a,b] and differentiable on (a,b) with f(a)=f(b), then there is a c from (a,b) such that f’(c)=0.

(2) Symbols succinctly encapsulate complicated ideas. Look at old mathematics and physics books and papers from the late 1800s — before much of the modern terminology was invented. They are a real pain to read because so much time is spent describing the ideas in long complicated sentences. I find myself reading those sentences over and over again to parse out the meaning. When I finally get it, it can usually be encapsulated in a few modern symbols.

7. 7. danielsdesk 5:25 pm 08/30/2013

@leeml I think you are missing the point of the article; They simply explain the symbols contribute to the complexity of math when it comes to those unfamiliar with them, but nowhere in the article are they saying math is better off without them. The symbols definitely do encapsulate much more than english language expression could, but you have to know the concept before you can understand seeing it encapsulated in short-form. The article is talking about how people perceive math, not that math should be without complex terms and symbols.

8. 8. jtdwyer 6:49 pm 08/30/2013

KBCash,
I guess that you must be a native English speaker. Unfortunately for native speakers of all other languages, most scientific journals are published only in English, even though some ideas are difficult to translate…

9. 9. jtdwyer 7:20 pm 08/30/2013

leeml,
Regarding your second point, that is precisely part of the problem as I see it: established mathematical formulas impart a sense of correctness that has not necessarily been established. For example, applying
M = (v^2 r)/G
to estimate the mass within the orbital radius of a planet works quite well to approximate the mass of the Sun, since 99.86% of total solar system mass is contained within the Sun. However, that same method commonly applied to ‘orbital’ objects within spiral galaxies does not properly estimate the total galactic mass distributed throughout the star’s orbit – it only determines the mass that would be necessary to produce the star’s velocity if it were all located at the specified radial distance (the galactic center). IMO, this and other contextual errors contribute to the perceived requirement for galactic dark matter to compensate for mass/gravitation discrepancies.

Let me know if this is not understandable – I can attempt alternative explanations, although I cannot produce what I think should be a relatively simple mathematical proof. As a result, my explanations are often dismissed out of hand…

BTW, I have little difficulty comprehending Newton’s ‘Principia’, despite the archaic spelling and language. I’m sure that someone much better versed in mathematics than I would prefer more efficient symbolic representations – but then I could not fully comprehend.

10. 10. gmperkins 11:05 pm 08/30/2013

The main thing I dislike in movies/TV is how they often portray ‘brilliance’ as total understanding of a deep subject without any effort. Like they were born understanding quantum mechanics. They never even hint at the years of study to get to a high level of understanding. Even the child prodigies spend years reading textbooks, they just start earlier than most. Aptitude reduces the time committment but it is always long. Yet movies/TV act like ‘if you don’t know it you never will because you weren’t born with it’. Really makes me angry, like a big excuse to stop thinking on anything/everything. I consider this a big problem in America where so many just use excuses to avoid a little mental effort to learn basic yet very useful math and science. Like kids who become mechanics, they suddenly realize way too late that just some basic math and physics would do wonders with their understanding the how and why of a car. I have seen this first hand it is a very bad cultural myth.

11. 11. kato_square 9:02 am 08/31/2013

Is there a confusion about intuition and precise reasoning?
The illustration of Rolle’s Theorem is what any instructor of an informal calculus class would give, and any instructor of a rigorous calculus would start with the same illustration, and then move to the formal statement. Both are useful — but only the formulation of Rolle’s Theorem, and its ilk, can be used for a full, correct development of the Calculus.

The article would have been improved with a informal description of why the birthday paradox works the way it does. It is a bit harder to draw a picture, and then again justifies the rigorous reasoning. Without it, the further hard and important applications of this beautiful paradox are very hard to find.

12. 12. jtdwyer 12:14 pm 08/31/2013

kato_square,
Actually, as I understand the birthday problem is not a proper paradox, as there is no apparently self-contradictory statement. I think it’s just a problem that is more complex than it appears at first, involving frequency distributions of random events. It’s crucially stipulated that birthdays are randomly distributed, thus reducing the probabilities of coincidence to a tractable mathematical problem.

Personally, I fail to see the beauty of pointless little puzzles, especially when one has to idealize the problem conditions in order to facilitate tractable solution!
Perhaps I’m missing something – what are the important applications of this idealized problem?

13. 13. burgerkinghorn 12:55 am 09/1/2013

“Mathematical notation, consisting of an array of letters, both Roman and Greek, with squiggly lines, and superscripts and subscripts, is one aspect of mathematics that intimidates the nonmathematician (and often some mathematicians, too). It is really a convenient way to relate complicated ideas to each other in a compact space. The “trick” in reading a mathematical paper is to recognize that each symbol has some meaning, to know the meaning when it is introduced, but then to take it on faith that you “understand” its meaning, and to pay attention to the way in which the symbol is manipulated. The essence of mathematical elegance is to produce a notation of symbols that is so simply organized that the reader can understand the relationships immediately.”
(Salburg, 2001, p. 230)

Salsburg, David (2001). The lady tasting tea: how statistics revolutionized science in the twentieth century. Owl Books

14. 14. Dr. Strangelove 11:25 pm 09/1/2013

@JTD
I’m sorry to say this whole business of Keplerian dynamics of yours is simply a lack of understanding of Newtonian mechanics and calculus. A long discussion with a professor of physics will clear it all up for you. No offense intended.

15. 15. Dr. Strangelove 11:35 pm 09/1/2013

Mathematical ideas are simple but mathematical proofs can be complex. It took thousands of years for somebody (Godel) to finally prove the simple idea that arithmetic is incomplete. Can you explain his proof in layman’s term?

How about the Monty hall problem? Even mathematicians are confused with it. It took Marilyn vos Savant, whose IQ is off-the-chart genius level, to explain to mathematicians the correct answer. In this case, even the idea is not simple.

16. 16. jtdwyer 9:37 am 09/2/2013

Dr. Strangelove,
No offense, but if you’ve identified my simple misunderstanding, then please, by all means provide some insight to your better understanding. Or do you merely presume that physics professors must surely understand my misunderstanding? Pardon me if I disregard your otherwise unexplained dismissal.

In fact, Vera Rubin concluded that there must be some missing mass _solely_ from her measurements of stellar rotational velocities plotted over radial distance, which did not diminish like planets in the Solar system. There was no analysis of Newtonian dynamics whatsoever!
It was the simple evidence of ‘flat’ galaxy rotation curves provided by those charts that convinced the astrophysical community of dark matter’s existence.

Plotting rotational velocity as a function of radial distance considers only that all gravitational potential originates at the axis of rotation. Recent problem statements now even recognize this misperception. See
http://en.wikipedia.org/wiki/Galaxy_rotation_curve
“The galaxy rotation problem is the discrepancy between observed galaxy rotation curves and the Newtonian-Keplerian prediction, assuming a centrally-dominated mass associated with the observed luminous material.”

Please see the seminal work, http://adsabs.harvard.edu/abs/1980ApJ…238..471R Section VIII. “DISCUSSION AND CONCLUSIONS” (page 485):
“1. Most galaxies exhibit rising rotational velocities at the last measured velocity; only for the very largest galaxies are the rotation curves flat. Thus the smallest Sc’s (i.e., lowest luminosity) exhibit the same lack of a Keplerian velocity decrease at large R as do the high-luminosity spirals. This form for the rotation curves implies that the mass is not centrally condensed, but that significant mass is located at large R. The integral mass is increasing at least as fast as R. The mass is not converging to a limiting mass at the edge of the optical image. The conclusion is inescapable that non-luminous matter exists beyond the optical galaxy.”

This cursory analysis ignores that the vast distribution of disk masses gravitationally interact with each other!

Unlike the peripheral masses within the Solar system, which primarily gravitationally interact only with the Sun (which contains 99.86% of total Solar system mass), peripheral spiral galaxy masses gravitationally interact with billions of comparably massive neighboring objects. The planar disks of spiral galaxies are principally self-gravitating – unlike the Solar system, whose gravitational potential originates within the Sun.

Newton proved in his ‘Principia’ that Kepler’s two-body equations provided useful approximations within the Solar system because the perturbations of other planets were so small that their influence is negligible in relation to the Sun. In spiral galaxies, it is the influence of central masses that is dominated by the ‘perturbations’ of hundreds of billions of discrete massive objects dispersed over many tens of thousands of light years. The source of gravitational potential cannot be properly treated as though it originates within the exceedingly remote galactic center.

This is more succinctly stated in http://arxiv.org/abs/1101.3224 – which applies the equations of relativistic dynamics to quite successfully describe spiral galaxy rotation:
“In dismissing general relativity in favour of Newtonian gravitational theory for the study of galactic dynamics, insufficient attention has been paid to the fact that the stars that compose the galaxies are essentially in motion under gravity alone (“gravitationally bound”).”
These authors simply dismiss the equations of Newtonian dynamics as inadequate, whereas I hold some hope that, when extended to properly represent the actual distribution of gravitational potentials, simpler solutions using Newtonian equations can also be applied. See http://arxiv.org/abs/1104.3236

Anyone, please explain my simple misunderstanding!

17. 17. Dr. Strangelove 11:28 pm 09/2/2013

JTD
I advise you to see a physics professor to explain to you how the center of mass of various objects are calculated. I assure you he will tell you Newton and Kepler got it right but you misunderstood their works due to your unfamiliarity with calculus.

As for Vera Rubin, arguing that there must be some missing mass based on observation does not mean Newtonian mechanics is wrong. Astronomers have attributed missing mass to dark matter.

Your great misunderstanding is believing deep in your heart that you discovered that Newtonian mechanics is wrong and does not apply to rotating galaxies, without bothering to learn Newtonian mechanics and calculus.

Professors who have written textbooks on Newtonian mechanics can tell you if it is wrong. You can trust them. They are not stupid. If they are wrong, universities would have stopped teaching Newtonian mechanics long ago.

18. 18. Dr. Strangelove 3:47 am 09/3/2013

JTD
It seems you don’t even understand the papers you are citing. None of the three papers plus Rubin suggest that the galactic mass is not at the center like the solar system as you keep saying.

Instead the papers and Rubin are arguing either there’s a missing mass (dark matter) or the gravitational force is not linearly proportional to acceleration. While the latter requires modifying Newtonian mechanics, it is not about the galactic center of mass you imagine it to be.

The Feng and Galo paper is just a mathematical trick. They arbitrarily inserted ‘galactic rotation numbers’ to match observations and calculations. Few physicists and astronomers support such mathematical tricks to change Newtonian mechanics. Dark matter is generally accepted by the science community because it does not require tricks.

19. 19. jtdwyer 7:14 am 09/3/2013

Dr. Strangelove,
The disks of spiral galaxies are collectively self-gravitating, not independent, non-interacting planets each effectively orbiting the Sun, or any collective center of mass.

Fundamentally, the characteristic rotational velocity diminishing with radial distance is a property of two-body interactions dominated by one of the two bodies.

Since the magnitude of the gravitational interaction does diminish greatly with distance, the motions (velocity) of peripheral disk objects, for example, are primarily determined not by the center of mass that is many tens of thousands of light years away, but by the billions of much nearer, comparably massive objects, which are also in motion.

The two referenced papers were both published in peer reviewed journals. The Feng & Gallo paper
http://arxiv.org/abs/1104.3236
attempts to use complex methods in applying Newtonian dynamics to properly represent the distribution of gravitational potential throughout galactic disks. Sorry if the mathematical methods they applied do not meet with you approval. I won’t attempt to explain further.

The Carrick and Cooperstock paper more naturally applies relativistic dynamics to describe the vast, self-gravitating disks.

Both papers papers represent earnest attempts to properly represent the gravitational interactions of the vast distributions of massive objects comprising the disks of spiral galaxies – rather than force fitting simply Keplerian planetary dynamics on vast, distributed, compound objects.

BTW – please refrain from putting words in my mouth that seem to fit your misunderstanding of the words I actually write. “None of the three papers plus Rubin suggest that the galactic mass is not at the center like the solar system as you keep saying.” I never said that. Feng & Gallo, for example, evaluate the distribution of mass and gravitation as a series of cylindrical partitions representing the measured velocities of test objects distributed throughout the disk. It is you who do not understand. Your personal insults are unfounded and incredulous. I have nothing more to say to you.

20. 20. jtdwyer 9:30 am 09/3/2013

I will complete the paragraph as intended:
The Carrick and Cooperstock paper http://arxiv.org/abs/1101.3224 more naturally applies relativistic dynamics to describe the vast galactic disks as self-gravitating fluid structures.

Neither paper determines the gravitational effects imparted to any portion of the disk as having originated solely from the galactic center (or center of mass, etc.). Both determine rotational velocities that are in close approximation to actual measurements – without relying on imaginary dark matter.

21. 21. polarise 6:59 am 09/5/2013

“Who wouldn’t be surprised to learn that in a room of 23 people, there is a 50% chance two of them share a birthday?”

The statement is incorrect. Even the website pointed to gets it right. It should read,

“Who wouldn’t be surprised to learn that in a room of 23 people, there is a 50% chance *at least* two of them share a birthday?”

22. 22. biggus56 8:39 am 09/5/2013

jtdwyer: you quote Vera Rubin twice, and damn yourself each time.
“If we observe the velocities of stars orbiting in the galaxy we find that their velocities remain flat all the way to the edge of our observations – that’s not what was expected.”
“Thus the smallest Sc’s (i.e., lowest luminosity) exhibit the same lack of a Keplerian velocity decrease at large R as do the high-luminosity spirals. This form for the rotation curves implies that the mass is not centrally condensed, but that significant mass is located at large R.”
Significant words: “all the way to the edge of our observations” and “at large R”.
An atomic nucleus seen from a distance appears pointlike. Up close, it’s a swirl of protons, neutrons, pions; closer still we get quarks, gluons. So intranuclear dynamics are highly complex; but move away and we can use Newtonian physics. When Rubin says “at large R”, she means “at large R”. And you’re wrong.

23. 23. jtdwyer 3:33 am 09/7/2013

Negatively charged electrons are bound to positively charged nuclei solely by the quantum electromagnetic interaction – this has nothing to do with gravitation.

What Rubin, et al., meant (in 1980) by large radii is illustrated in figure 6 on page 480 in the referenced paper (see link in comment 16). Their use of the term “large R” is relative to the peripheral stars whose velocities they measured in each galaxy – ranging from about 7 kpc for the smaller spiral galaxies to more than 25 kpc for the largest. I have no idea what imaginary “significance” you attributed to those words!

A much more recent study of 46 distant stars that orbit our galaxy outside the disk (at radial distances ranging from 50 to 150 kpc) and not aligned with it, forming the stellar halo, shows that they _do_ comply with Keplerian dynamics – their velocities diminish as a function of radial distance. The study actually uses the disparity in their velocities to estimate the galactic mass within their orbits – finding that 4/5 of the galactic mass is located with 50 kpc of the galactic center. This is far smaller than the enormous mass thought to lie far outside the observable galaxy periphery – in the form of a giant dark matter halo.
Also note that these stars within the stellar halo each independently orbit the distant galaxy bulge and disk, while objects within the galactic disk are self-gravitating (collectively interacting – bound to each other) – they do not independently orbit any galactic center. This is why the halo stars’ rotational velocities diminish as a function of their radial distance – they are independently orbiting the galaxy much like planets, in effect, independently orbit the Sun.
See http://arxiv.org/abs/1205.6203. Also see http://arxiv.org/abs/1108.1629.

24. 24. jtdwyer 7:26 am 09/16/2013

Regarding my comment #3 and referenced essay, please see a report of a remarkably similar argument, albeit a more scholarly one – http://www.siam.org/pdf/news/2094.pdf

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